线性代数课业代做线性代数代写

Problems

线性代数课业代做 Instructions 1.Supply complete, rigorous solutions to each of the problems below.2.Cite the result or number when using a nontrivial

Instructions

Supply complete, rigorous solutions to each of the problems below.
Cite the result or number when using a nontrivial lemma/proposition/theorem from class or the readings.
External textbooks or websites are not permitted.

1.True or false?

(a) A linear combination of surjective mappings V → V is again surjective.

(b) A linear combination of injective mappings V → V is again injective.

(c) If A,B are two linear mappings V → V, then AB = I implies BA = I. (Hint: V is allowed to be infinite-dimensional here.)

2. 线性代数课业代做

Recall what it means for two linear mappings A,B : V →V to be similar. If S is a set of linear mappings (or matrices), then S is similarity invariant if A ∈ S and A ∼ B implies B ∈ S.

Which of the following sets is similarity invariant?

(a) The set of invertible n×n matrices

(b) The set of diagonal n×n matrices

(d) The set of nilpotent linear mappings V → V

(e) The set of symmetric n×n matrices

(f) The set of skew-symmetric n×n matrices

(g) The set of n×n matrices of finite order; i.e. those matrices such that A^k = I for some k.

(h) The set of n×n matrices with determinant equal to 1

Let V be a vector space. Prove that dim(V) < ∞ if and only if AB = I =⇒ BA = I for all linear mappings A,B : V → V.

5.Sums of subspaces.

(a) Let W₁,W₂ be two subspaces of a vector space V. Show that the union W₁∪W₂ may not be a subspace; in fact show that W₁∪W₂ is a subspace if and only W₁ ⊆ W₂ or W₂ ⊆ W₁.

Since the union of two subspaces isn’t a subspace, we need a different way to “combine” two subspaces …

(b) Complete the definition: the subspace sum of W₁ and W₂ is the set W₁ +W₂ = {··· }.

(d) In R³ , let W := span{(1,0,−1)}. Find a subspace W′ such that R³ = W +W′.

(e) Let B1,B2 be bases for W₁,W₂. Show that B₁∪B₂ is a spanning set for W₁ +W₂, but may not be a basis.

(f) If U,W,X are three subspaces such that U + X = W + X, does it follow that U = W? Proof or counterexample.

(g) Suppose that V is finite dimensional and W₁,W₂ are subspaces of V. Find a formula relating the dimensions of W₁ +W₂, W₁, W₂, and W₁∩W₂.

6. 线性代数课业代做

We write V = W₁⊕W₂to mean that V = W₁+W₂ and W₁∩W₂ = {0}. This is called a direct sum.

(a) Prove that the following are equivalent for subspaces W₁,W₂of a vector space V.

(i) W₁∩W₂ = {0}.

(ii) If w₁ ∈ W₁ and w₂ ∈ W₂ are nonzero vectors, then {w₁,w₂} is linearly independent.

(iii) If w₁ +w₂ = 0 for some w₁ ∈ W₁ and w₂ ∈ W₂, then w₁ = w₂ = 0.

(b) In R³ , let W := span{(2,1,0),(0,1,−1)}. Find a subspace W′ such that R³ = W⊕W′.

(c) Recall the following definitions: transpose of a matrix, symmetric matrix, and skew-symmetric matrix. Let Sym_n(K) and SkewSym_n(K) be the sets of symmetric and skew-symmetric n × n matrices over K, respectively. Show that

M_n(K) = Sym_n(K)⊕SkewSym_n(K).

(d) Suppose V = W₁⊕W₂. Show that if B₁,B₂ are bases for W₁,W₂, then B₁ ∪B₂ is a basis for V. 线性代数课业代做

This shows why it’s useful to break a vector space into a direct sum: it makes it easy to find a basis for V. Just find a basis for each part, then take the union to find a basis for V.

(e) Let V be finite-dimensional and suppose V = W₁ +W₂. Show that V = W₁⊕W₂ if and only if dim(W₁) +dim(W₂) = dim(V).

(f) Is (e) true if we don’t originally assume that V = W₁ +W₂? i.e. is the following statement true: “if W₁,W₂ are subspaces of V, then V = W₁⊕W₂ if and only if dim(W₁) +dim(W₂) = dim(V)”?

8.Image and nullspace review. 线性代数课业代做

(a) Complete the definition: if T : V → V is a linear mapping, the image of T is the set

T(V) = im(T) = {··· }.

(b) Prove that im(T) is a subspace of V.

ker(T) = null(T) = {··· }.

(d) Prove that ker(T) is a subspace of V.

(e) Prove that T is injective if and only if ker(T) = 0.

(f) Let T : V → V be any linear mapping. Prove that V = ker(T)⊕im(T).

9.Projections. 线性代数课业代做

These will be very important. A linear mapping P : V → V is a projection if P²= P.

(a) Give three different examples of projections R² → R² .

(b) Let n ≥ 1. For which k ∈ N does there exist a projection P of rank k? Give a comprehensive answer.

(c) Pete disagrees with your answer in part (a); he can show that in finite dimensions, the only projections are 0 and I. First, Pete observes that in the finite dimensional case we can assume that V is isomorphic to Kⁿ and we can think of linear mappings as n×n matrices. So if P is an n×n matrix such that P² = P, then P² − P = 0, which factorizes as P(I − P) = 0. But this implies P = 0 or P = I, because a product of matrices is zero only if one of them is zero.

What is wrong with Pete’s proof?

10.

(a) Let W be any subspace of V. Show that there is always a projection P :V →V such that im(P) =W; thus P called a projection onto W. (Hint: find a basis for W, then extend it to V …)

(b) Let P,Q : V → be two projections. Is it true that if im(P) = im(Q), then P = Q?

14.Let T : V → V be a linear mapping and let W be a subspace. Let P : V → V be a projection onto W, i.e. P²= P and im(P) = W.

Show that W is T-invariant if and only if T P = PT P.

16.Eigenvalues II.

(a) Let P₃(R) be the space of polynomials of degree ≤ 3 with real coefficients, and let T : P₃(R) → P₃(R) be the differentiation operator. Find all eigenvalues of T.

(b) Same as (a), except with R replaced by C.

17.Eigenvalues III.

(a) Let T : V → V be a linear mapping. Show that λ is an eigenvalue for T if and only if the linear mapping T −λI is not injective.

(b) Show that T : V →V has an eigenvalue if and only if there is a 1-dimensional T-invariant subspace W of V.

(c) Show that if V is a finite-dimensional vector space over a field K, then every linear mapping T : V → V has at most finitely many eigenvalues. But show that there is an infinite-dimensional real vector space V and a linear mapping T : V → V which has every real number as an eigenvalue. (Hint for the second part: e^cx.)

20.Let V be a finite-dimensional vector space and let T : V → V be a linear mapping. Show that if T^k= I for some k ∈ N, then T is diagonalizable.

21.Let T : V → W be a linear mapping.

(a) Show that T is surjective if and only if T has a right inverse, i.e. there is a linear map S : W → V such that T S = I.

(b) Show that T is injective if and only if T has a left inverse, i.e. there is a linear map S : W →V such that ST = I.

(d) Give a counterexample to show that (c) fails when V is infinite-dimensional; a good example is the vector space V = K N of infinite sequences with elements in K.

22.True or false? 线性代数课业代做

Let f : U → V and g : V → W be linear mappings.

(a) If f,g are isomorphisms, then so is g ◦ f .

(b) If g ◦ f is an isomorphism, then so are both f and g.

(d) If g ◦ f is injective, then so are both f and g.

(e) If f,g are surjective, then so is g ◦ f .

(f) If g ◦ f is surjective, then so are both f and g.

23.Let K be an uncountable field. Construct an uncountable set of matrices S ⊆ M_n(K) such that AB = BA for all A,B ∈ S and such that A²= I for all A ∈ S. Or, prove that no such set exists! What happens if K is countable (g. K = Q)?

24.Show that every invertible matrix has a square root. More precisely: if A ∈ Mn(K) is an invertible matrix, then there is a matrix B ∈ M_n(K) such that B²= A. (Assume K is algebraically closed and char(K) ≠2.)

25.Let A,B be square matrices of the same size. Show that if I − AB is invertible, then I − BA is also invertible.

26.Linear recurrences. 线性代数课业代做

This one is fun! Let K be a field and let K^Ndenote the set of functions x : N → K; we think of these functions as sequences of elements of K. We say that x ∈ K^Nis a linear recurrence if there exists d ≥ 1 and scalars c₁,…, c_d ∈ K such that the following relation holds for all n ≥ d:

x(n) = c₁x(n−1) +···+c_dx(n−d).

The simplest example of a linear recurrence is the Fibonacci sequence, which satisfies the recurrence relation x(n) = x(n−1) +x(n−2) (thus d = 2 for the Fibonacci sequence).

Let Lin(K) denote the set of linear recurrences x : N → K. In this problem, we will show that Lin(K) is a subspace of K^N, which is a lot harder than it seems!

(a) Show that K^N is a vector space over K (with pointwise operations).

(b) Let S : K^N → K^N be the shift map given by 线性代数课业代做

S(x)(n) := x(n+1).

Show that S is a linear mapping. Is it injective? Is it surjective?

(d) For x ∈ K^N, define a subspace V(x) of K^N by

V(x) := span_K{x,S(x),S²(x),…}.

Show that V(x) is S-invariant. Show that x is a linear recurrence if and only if V(x) is finite-dimensional.

(e) Let x ∈ K^N. Show that x is a linear recurrence if and only if there is an S-invariant subspace W of K^N such that x ∈ W.

(f) Use (e) to prove that Lin(K) is a subspace of K^N; thus a linear combination of linear recurrences is again a linear recurrence.

27.

Is every monic polynomial the characteristic polynomial of some matrix? If two matrices have the same characteristic polynomial, must the matrices be similar?

33.Can you find a 5×5 nilpotent matrix of nilpotency index 6? In general, can you find an n×n nilpotent matrix of nilpotency index k, where k > n?

34.Let A,B be commuting nilpotents. Prove that A+B and AB are both nilpotent.

35.

Let Nil(V) denote the set of nilpotent transformations V → V and let GL(V) be the set of of invertible transformations V → V.

(a) Is Nil(V) a subspace of End(V)? Is Nil(V) closed under composition?

(b) Is GL(V) a subspace of End(V)? Is GL(V) closed under composition?

36.Let A and B be two commuting nilpotents. Prove that A+B and AB are both nilpotent.

37. 线性代数课业代做

Show that a nilpotent transformation can only have 0 as an eigenvalue. (Hint: If T(v) = λv, then T²(v) = λ²v …)

38.(a) Let T : V → V be a transformation of a vector space V over an algebraically closed field K. Prove that if the only eigenvalue of T in K is zero, then T is nilpotent. (Hint: what kind of matrix will you get if you triangularize?)

(b) Is (a) true if K is not algebraically closed? Proof or counterexample.

39.Let T : V → V be a linear mapping of an n-dimensional vector space.

(a) Show that if T is triangularizable, then T has invariant subspaces of every dimension: i.e. there are T-invariant subspaces W₁,…,W_n of V so that dim(W_i) = i for all 1 ≤ i ≤ n. (Notice that this appears to be weaker than the existence of a flag.)

(b) Is the converse of (a) true? Proof or counterexample.

40. 线性代数课业代做

Let T : V → V be a linear mapping of a finite-dimensional vector space V, and let W be a T-invariant subspace of V. Let T|_W: W → W be the restriction of T to W. Prove that the characteristic polynomial of T|_W divides the characteristic polynomial of T.

41.Let S,T : V → V be two linear mappings, and let W be a subspace of V which is both S-invariant and T-invariant. Show that W is invariant for both S+T and ST.

42.Let V be a finite-dimensional vector space.

(a) Let W₁,W₂ be two subspaces of V. Fill in the following formula (with proof):

dim(W₁ +W₂) = dim(W₁) +dim(W₂)−[something]

(b) Find a similar formula for three subspaces:

dim(W₁ +W₂ +W₃) = ···

45.

Let V be finite-dimensional. Let T : V →V be a linear mapping such that Tⁿ= I for some n ≥ 1; we say that T is a mapping of finite order. Show that W is a T-invariant subspace of V, then V = W⊕W′ for some T-invariant subspace W′ .

(You might want to come back to this after we learn inner products.)

46.Let T : V → V be a linear mapping and let W be a T-invariant subspace of V. Prove that if T is triangu-larizable then so is T|_W.

47.Let U,N ∈ M_n(K) be matrices where U is invertible and N is nilpotent. It is well-known that U +N is guaranteed to be invertible. Or wait, was it nilpotent? Shoot, I forgot. Figure it out and supply a proof.

48.Understanding cyclic vectors. 线性代数课业代做

Let T :V →V be a linear map and let x ∈V. Let Z be the cyclic subspace generated by x:

Z(x) = span{x,T(x),T²(x),…,}.

(a) Show that Z(x) is T-invariant.

(b) Show that there exists k ≥ 1 so that B = {x,T(x),…,T^k(x)} is a basis for Z(x).

(c) Let B be as in (b). Suppose T^k⁺¹(x) = 0; what’s nice about [T|_Z₍_x₎ ]_B? Even if T^k⁺¹(x) ≠ 0, the matrix [T|_Z₍_x₎ ]_B is still pretty nice — why?

49.Let T : V → V with V finite-dimensional.

50.Let T : V → V with V finite-dimensional. Suppose Tr(Tⁱ) = 0 for all i. Prove that T is nilpotent.

51.

Let A,B ∈ M_n(K). Let us write A ∼_KB to mean that there is an invertible matrix P ∈ M_n(K) such that PAP⁻¹= B; you can prove that ∼_K is an equivalence relation. Determine, with proof, which of the following statements is true.

(a) Let A,B ∈ M_n(R). If A ∼_R B then A ∼_C B.

(b) Let A,B ∈ M_n(R). If A ∼_C B then A ∼_R B.

(d) Let A,B ∈ M_n(Q). If A ∼_R B then A ∼_Q B.

52.(a) Let A ∈ M₃(Q) be a 3×3 matrix with eigenvalues λ = 1,2,3. Express A⁻¹ as a linear combination of powers of A.

(b) More generally: let T : V → V be a linear map with dim(V) < ∞. Show that if T is invertible, then T⁻¹ is a linear combination of powers of T.

53.Division algorithm for polynomials. 线性代数课业代做

Let f(x),g(x) ∈ F[x] be two polynomials over a field F with g(x) ≠0. Show that there exist polynomials q(x),r(x) ∈ F[x] such that

f(x) = g(x)q(x) +r(x)

and deg(r(x)) < deg(g(x)). (Note: the degree of a constant polynomial is zero, with the exception of the zero polynomial; by convention, the degree of the zero polynomial is −∞!)

The strategy is remarkably identical to the proof used for integers. Define a set of polynomials S by

S = {r(x) ∈ F[x] : there exists q(x) ∈ F[x] such that r(x) = f(x)−g(x)q(x)}.

Then S is nonempty. So by well-ordering, we can select an element r(x) ∈ S of least degree. Now argue that this r(x) is as required.

58.

Let T : V → V be a linear mapping of a finite-dimensional vector space V. Let W_i= ker(Tⁱ).

(a) Show that we have a decreasing chain of subspaces W₁ ⊇ W₂ ⊇ W₃ ⊇ ···.

(b) Show that the chain stabilizes: there is some i ∈ N so that W_i = W_i₊₁ = W_i₊₂ = ···. In fact show that we can take i ≤ dim(V).

(c) [Maryam] True or false? If W_i = W_i₊₁ for some i, then W_i = W_i₊_d for all d ≥ 1. (In plain language: if the chain stabilizes at some point, then the entire chain terminates at that point.)

59.Let F₂= {0,1} be the field of two elements. Here we’ll use polynomials and matrices to construct a field with four elements.

(a) Give an example of a 2×2 matrix A ∈ M₂(F₂) whose minimal polynomial is x² +x+1.

(b) Let F = {0,I,A,A²}. Show that F is a field (when equipped with usual matrix addition and multiplication).

60. 线性代数课业代做

A matrix A ∈ M_n(F) has finite order if A^k= I for some k ∈ N. The smallest such integer k is called the order of A (not dissimilar from the definition of nilpotency index).

(a) True or false? If A is a matrix of order k, then m_A(x) = x^k −1.

(b) True or false? Every matrix of finite order is diagonalizable.

61.

Let L(V) denote the space of linear transformations T : V → V. For S,T ∈ L(V), the Lie bracket of S and T is

[S,T] := ST −T S.

For T ∈ L(V), define a linear mapping ad(T) : L(V) → L(V) by

ad(T)(S) = [T,S].

This is called the adjoint representation of T.

(a) True or false? If ad(T) = ad(S), then T = S.

(b) Show that ad : L(V) → L(L(V)) is a linear mapping.

(d) Prove the product rule: ad(T)([X,Y]) = [ad(T)(X),Y] +[X, ad(T)(Y)].

(e) True or false? If T is nilpotent, then ad(T) is nilpotent.

(f) True or false? If T is diagonalizable, then ad(T) is diagonalizable.

(g) Show that if T = D+N is the Jordan–Chevalley decomposition of T, then ad(T) = ad(D) +ad(N) is the Jordan–Chevalley decomposition of ad(T).

62. 线性代数课业代做

Let W be a subspace of V. A complement for W is a subspace W′ such that V = W⊕W′ .

(a) Show that every subspace has a complement. Is the complement necessarily unique?

(b) Suppose V is finite-dimensional over an algebraically closed field F, and let T : V → V be a linear mapping. Prove that T is diagonalizable if and only if every T-invariant subspace of V has a T-invariant complement.

63. 线性代数课业代做

Let T : V → V be a linear mapping of a finite-dimensional vector space V and let W be a T-invariant subspace of V. If T|_Wis diagonalizable, does it follow that T is diagonalizable? What if we replace “diagonalizable” with“triangularizable”?

64.A linear mapping T : V → V satisfies the following criteria. Deduce, with proof, the Jordan form of T and the corresponding dot diagram.

p_T(x) = x²(x−1)³(x−2)⁴ , m_T(x) = x(x−1)²(x−2)² , dim(E₂) = 2.

65.Let T : V → V be a linear mapping with dim(V) = 7, rank(T) = 2, and suppose that T has exactly three distinct eigenvalues. Prove that T is diagonalizable.

66.Find, with proof, all n ∈ N for which the following statement is true.

If A,B ∈ M_n(C) are two nilpotent matrices with the same rank and same nilpotency index, then A ∼ B.

69.Preduals. Let V be an F-vector space. A predual for V is an F-vector space W such that W∗ ≃ V.

Which of the following vector spaces have preduals?

R R²

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